Higher June 2018 Paper 1 Q20
20 \(n\) is an integer such that \(3n + 2 \leqslant 14\) and \(\dfrac{6n}{n^2 + 5} \gt 1\)
Find all the possible values of \(n\). (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| 2, 3, 4 | M1 | for method to solve \(3n + 2 \leqslant 14\) eg \(n \leqslant (14 - 2) \div 3\) oe |
| M1 | for complete method to rearrange \(\dfrac{6n}{n^2 + 5} \gt 1\) to the form \(an^2 + bn + c\ (\lt 0)\) | |
| M1 | for method to begin to solve \(n^2 - 6n + 5\ (\lt 0)\) eg \((n \pm 5)(n \pm 1)\ (\lt 0)\) | |
| M1 | (dep on previous M2) for \(n \gt 1\) and \(n \leqslant 4\) or \(1 \lt n \lt 5\) | |
| A1 | (dep M4) cao |
Additional guidance
This could be shown within an equation rather than an inequality at this stage
For the 2nd and 3rd M marks condone no ‘\(\lt 0\)’ and condone use of incorrect inequality signs or ‘=’
Accept \(\dfrac{-(-6) \pm \sqrt{(-6)^2 - 4 \times 1 \times 5}}{2 \times 1}\) (condone one sign error)
Must come from correct working
Could be shown on a number line
Alternative method
| Answer | Mark | Mark scheme |
|---|---|---|
| 2, 3, 4 | M1 | for method to solve \(3n + 2 \leqslant 14\) eg \(n \leqslant (14 - 2) \div 3\) oe OR for \(3 \times 4 + 2 = 14\) |
| M3 | for trials with 1, 2, 3 and 4 in the quadratic inequality, correctly evaluated | |
| (M2 | for trials with three of 1, 2, 3 and 4, correctly evaluated) | |
| (M1 | for trials with two of 1, 2, 3 and 4, correctly evaluated) | |
| A1 | (dep M4) cao |
This could be shown within an equation rather than an inequality at this stage
The values from the trials may be given as improper fractions eg \(\dfrac{24}{21}\), \(\dfrac{18}{14}\), \(\dfrac{12}{9}\), \(\dfrac{6}{6}\)